Abstract
Is Winfree’s abstract Tile Assembly Model (aTAM) “powerful?” Well, if certain tiles are required to “cooperate” in order to be able to bind to a growing tile assembly (a.k.a., temperature 2 self-assembly), then Turing universal computation and the efficient self-assembly of N ×N squares is achievable in the aTAM (Rotemund and Winfree, STOC 2000). So yes, in a computational sense, the aTAM is quite powerful! However, if one completely removes this cooperativity condition (a.k.a., temperature 1 self-assembly), then the computational “power” of the aTAM (i.e., its ability to support Turing universal computation and the efficient self-assembly of N ×N squares) becomes unknown. On the plus side, the aTAM, at temperature 1, is not only Turing universal but also supports the efficient self-assembly N ×N squares if self-assembly is allowed to utilize three spatial dimensions (Fu, Schweller and Cook, SODA 2011). In this paper, we investigate the theoretical “power” of a seemingly simple, restrictive variant of Winfree’s aTAM in which (1) the absolute value of every glue strength is 1, (2) there is a single negative strength glue type and (3) unequal glues cannot interact (i.e., glue functions must be “diagonal”). We call this abstract model of self-assembly the restricted glue Tile Assembly Model (rgTAM). We achieve two positive results. First, we show that the tile complexity of uniquely producing an N ×N square in the rgTAM is O(logN). In our second result, we prove that the rgTAM is Turing universal.
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Patitz, M.J., Schweller, R.T., Summers, S.M. (2011). Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue. In: Cardelli, L., Shih, W. (eds) DNA Computing and Molecular Programming. DNA 2011. Lecture Notes in Computer Science, vol 6937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23638-9_15
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DOI: https://doi.org/10.1007/978-3-642-23638-9_15
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